3.12.80 \(\int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx\)

Optimal. Leaf size=90 \[ -\frac {277750}{3 x+2}-\frac {75625}{5 x+3}-\frac {46475}{2 (3 x+2)^2}-\frac {7480}{3 (3 x+2)^3}-\frac {1133}{4 (3 x+2)^4}-\frac {154}{5 (3 x+2)^5}-\frac {49}{18 (3 x+2)^6}+1615625 \log (3 x+2)-1615625 \log (5 x+3) \]

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Rubi [A]  time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} -\frac {277750}{3 x+2}-\frac {75625}{5 x+3}-\frac {46475}{2 (3 x+2)^2}-\frac {7480}{3 (3 x+2)^3}-\frac {1133}{4 (3 x+2)^4}-\frac {154}{5 (3 x+2)^5}-\frac {49}{18 (3 x+2)^6}+1615625 \log (3 x+2)-1615625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/(3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^
2) - 277750/(2 + 3*x) - 75625/(3 + 5*x) + 1615625*Log[2 + 3*x] - 1615625*Log[3 + 5*x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx &=\int \left (\frac {49}{(2+3 x)^7}+\frac {462}{(2+3 x)^6}+\frac {3399}{(2+3 x)^5}+\frac {22440}{(2+3 x)^4}+\frac {139425}{(2+3 x)^3}+\frac {833250}{(2+3 x)^2}+\frac {4846875}{2+3 x}+\frac {378125}{(3+5 x)^2}-\frac {8078125}{3+5 x}\right ) \, dx\\ &=-\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (2+3 x)-1615625 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 92, normalized size = 1.02 \begin {gather*} -\frac {277750}{3 x+2}-\frac {75625}{5 x+3}-\frac {46475}{2 (3 x+2)^2}-\frac {7480}{3 (3 x+2)^3}-\frac {1133}{4 (3 x+2)^4}-\frac {154}{5 (3 x+2)^5}-\frac {49}{18 (3 x+2)^6}+1615625 \log (5 (3 x+2))-1615625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/(3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^
2) - 277750/(2 + 3*x) - 75625/(3 + 5*x) + 1615625*Log[5*(2 + 3*x)] - 1615625*Log[3 + 5*x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2), x]

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fricas [A]  time = 1.18, size = 155, normalized size = 1.72 \begin {gather*} -\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 408159415125*x^3 + 202262350455*x^2 + 29081250
0*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 29081250
0*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(3*x + 2) + 53445037
346*x + 5882909754)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)

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giac [A]  time = 0.85, size = 85, normalized size = 0.94 \begin {gather*} -\frac {75625}{5 \, x + 3} + \frac {625 \, {\left (\frac {22074930}{5 \, x + 3} + \frac {16294797}{{\left (5 \, x + 3\right )}^{2}} + \frac {6120660}{{\left (5 \, x + 3\right )}^{3}} + \frac {1179210}{{\left (5 \, x + 3\right )}^{4}} + \frac {94660}{{\left (5 \, x + 3\right )}^{5}} + 12117357\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 1615625 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="giac")

[Out]

-75625/(5*x + 3) + 625/4*(22074930/(5*x + 3) + 16294797/(5*x + 3)^2 + 6120660/(5*x + 3)^3 + 1179210/(5*x + 3)^
4 + 94660/(5*x + 3)^5 + 12117357)/(1/(5*x + 3) + 3)^6 + 1615625*log(abs(-1/(5*x + 3) - 3))

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maple [A]  time = 0.01, size = 81, normalized size = 0.90 \begin {gather*} 1615625 \ln \left (3 x +2\right )-1615625 \ln \left (5 x +3\right )-\frac {49}{18 \left (3 x +2\right )^{6}}-\frac {154}{5 \left (3 x +2\right )^{5}}-\frac {1133}{4 \left (3 x +2\right )^{4}}-\frac {7480}{3 \left (3 x +2\right )^{3}}-\frac {46475}{2 \left (3 x +2\right )^{2}}-\frac {277750}{3 x +2}-\frac {75625}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2/(3*x+2)^7/(5*x+3)^2,x)

[Out]

-49/18/(3*x+2)^6-154/5/(3*x+2)^5-1133/4/(3*x+2)^4-7480/3/(3*x+2)^3-46475/2/(3*x+2)^2-277750/(3*x+2)-75625/(5*x
+3)+1615625*ln(3*x+2)-1615625*ln(5*x+3)

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maxima [A]  time = 0.63, size = 86, normalized size = 0.96 \begin {gather*} -\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 1615625 \, \log \left (5 \, x + 3\right ) + 1615625 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 408159415125*x^3 + 202262350455*x^2 + 53445037
346*x + 5882909754)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) - 161
5625*log(5*x + 3) + 1615625*log(3*x + 2)

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mupad [B]  time = 1.09, size = 76, normalized size = 0.84 \begin {gather*} 3231250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {323125\,x^6}{3}+\frac {7690375\,x^5}{18}+\frac {114373325\,x^4}{162}+\frac {67186735\,x^3}{108}+\frac {499413211\,x^2}{1620}+\frac {26722518673\,x}{328050}+\frac {980484959}{109350}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2/((3*x + 2)^7*(5*x + 3)^2),x)

[Out]

3231250*atanh(30*x + 19) - ((26722518673*x)/328050 + (499413211*x^2)/1620 + (67186735*x^3)/108 + (114373325*x^
4)/162 + (7690375*x^5)/18 + (323125*x^6)/3 + 980484959/109350)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 +
(268*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)

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sympy [A]  time = 0.22, size = 83, normalized size = 0.92 \begin {gather*} \frac {- 70667437500 x^{6} - 280314168750 x^{5} - 463211966250 x^{4} - 408159415125 x^{3} - 202262350455 x^{2} - 53445037346 x - 5882909754}{656100 x^{7} + 3018060 x^{6} + 5948640 x^{5} + 6512400 x^{4} + 4276800 x^{3} + 1684800 x^{2} + 368640 x + 34560} - 1615625 \log {\left (x + \frac {3}{5} \right )} + 1615625 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2/(2+3*x)**7/(3+5*x)**2,x)

[Out]

(-70667437500*x**6 - 280314168750*x**5 - 463211966250*x**4 - 408159415125*x**3 - 202262350455*x**2 - 534450373
46*x - 5882909754)/(656100*x**7 + 3018060*x**6 + 5948640*x**5 + 6512400*x**4 + 4276800*x**3 + 1684800*x**2 + 3
68640*x + 34560) - 1615625*log(x + 3/5) + 1615625*log(x + 2/3)

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